What is known about continuum percolation in 1d?
By this, I mean, for $d \in \mathbb{N}$, the Poisson-Boolean model of disks of radius $r_0 \in \mathbb{R}$ with centres arranged randomly in $[0,1]^{d}...

Define the planar percolation where each unit edge is open with probability $p$ very close to $1$.
Looking at the event where there exists a directed open path between $(0,0)$ and $(n,n)$. This event ...

Let $G$ be the $n$-dimensional boolean hypercube, i.e. the graph on $\{0,1\}^n$ where two vertices are adjacent iff they differ on exactly one coordinate. Consider a graph $G'$ obtained by deleting a ...

Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...

I am trying to compute the equilibrium measure for the Meixner ensemble on page 19 (on the arxiv version). The "details" of the computation are in Section 6, where he finds the equilibrium measure is ...

In the context of percolation, e.g., bond/site percolation, random graph connectivity in 2-3 dimensions, etc., once the percolation threshold is reached, that is the system is spanned by an infinite ...

I read the Wikipedia article on Percolation critical exponents. It says:
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of ...

Let $G$ be the $n \times n$ grid, in which each vertex is connected to the vertices above it, below it and on either side. Let $G_p$ be the random subgraph of $G$ obtained by keeping each edge with ...

Let $G$ be an acyclic directed graph with $MN$ vertices arranged into $M$ generations of $N$ vertices each. We stipulate that edges may only go from generation $j$ to generation $j+1$, so there are $(...

This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...

Consider Hex on an $n \times n$ board without a swap rule, so that the first player wins. Assume the first player tries to minimize the length of the game, and the second player tries to maximize the ...

This is a sequel to the question: Proof and interpretation of the following percolation theory result for $n\times n$ square grid
In the paper: The Birth of the Infinite Cluster:Finite-Size Scaling ...

As part of my graduate research I am currently studying the last section in the paper "Random Majority Percolation" by Balister, Bollobas et al. The paper itself is very complicated but the last two ...

I was looking at this paper “Islands in Sea” and “Lakes in Mainland” phases and related transitions simulated on a square lattice on Percolation theory. The concept of phase transition used here seems ...

Consider a rooted tree of height $h$, such that all the leaves are at last layer. We perform the following random process: each edge is deleted with probability $0.5$, and otherwise it is retained. We ...

One of the weakest estimates conjectured to hold for critical planar percolation models (and proved in many cases) is the so-called RSW estimate. RSW estimate is the statement that the probability of ...

Consider a graph $G$ with a particular node $i$ labeled as “infected”. Other nodes start uninfected, and will become infected over time according to the following process: To each edge of the graph, ...

Suppose we have a shape bounded by a simple closed curve $\gamma \subset \mathbb{C}$, with points $A,B,C,D$ in cyclic order on the curve.
If we randomly color the interior of that shape in half red ...

Kindly refer to this paper: https://arxiv.org/abs/1502.03050
In this paper, Hugo Duminil-Copin and Vincent Tassion have given an alternative proof of the well known results. I was reading this paper ...

I am stuck in a part of my research which I am not expert in.
I have a 2-dimensional square lattice with periodic boundary conditions(torus). I am placing one walker at each node at the beginning. It ...

I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic ...